A special window has been added in Mathgrapher v2 to allow detailed presentation
of 2D orbits at the pixel level and to study the stability of the orbits (see the Examples and Demonstrations for the Henon map, Standard map and Mandelbrot and Julia sets.
In the second half of the last century it was discovered that most (non-linear) dynamical
systems exhibit choatic behaviour. This came as a surprise. It was thought that
deterministic systems would behave in a predictable way and that choatic behaviour
in such systems would be due to many different external influences on a system. It
turned out, however, that non-linear systems which are fully deterministic and quite
simple and straightforward in mathematical sense often show chaotic behaviour. This
means that a very small change in the initial conditions of the system may may lead
to completely different end results thereby making the outcome completely unpredictable.
This was surprising because it occurs in systems whose behaviour is governed by
well-understood physical principles described by exact mathematical formulae. The
example that is often mentioned is the butterfly in Alaska which may cause a
thunderstorm in Texas just by flapping its wings.
Scientist may be surprised by this finding, however every pinball player knows
that a very slight change in the velocity of the ball at some point along its
trajectory (for instance by kicking the machine) may change the end result completely.
The examples contain two well-known examples of systems that exhibit chaotic behaviour.
Both are incorporated in the Demonstrations, so you just have to choose Demonstrations
from Mathgrapher's menu to see how you may
calculate and analyse such systems. The first example is a simple iterative
algorithm: the logistic map. The second example is a well known system of
Ordinary Differential Equations (ODE's): the Lorenz equations.